Hamilton's theory of kin selection is the best-known framework for understanding the evolution of social behavior but has long been a source of controversy in evolutionary biology. A recent critique of the theory by Nowak, Tarnita, and Wilson sparked a new round of debate, which shows no signs of abating. In this overview, we highlight a number of conceptual issues that lie at the heart of the current debate. We begin by emphasizing that there are various alternative formulations of Hamilton's rule, including a general version, which is always true; an approximate version, which assumes weak selection; and a special version, which demands other restrictive assumptions. We then examine the relationship between the neighbor-modulated fitness and inclusive fitness approaches to kin selection. Finally, we consider the often-strained relationship between the theories of kin and multilevel selection.

The pithiest expression of the concept of kin selection was made long before the theory itself was devised, when J. B. S. Haldane is said to have quipped, “I would lay down my life for two brothers or eight cousins.” The remark captures an intuitive and powerful thought: When interacting organisms share genes, they may have an evolutionary incentive to help each other. Moreover, and more profoundly, it suggests that the size of the incentive to help is proportional to the degree of relatedness between them. We owe the formal embodiment of this insight to Hamilton (1964) and the term kin selection to Maynard Smith (1964). Today, Hamilton's (1964) theory lies at the heart of an established and sizeable research program, the explanatory domain of which has steadily expanded (Bourke 2011a).

The basic empirical prediction of kin selection theory is that social behavior should correlate with genetic relatedness; in particular, altruistic actions, which are costly to the actor but benefit others, are more likely to be directed toward relatives. This qualitative prediction has been amply confirmed in diverse taxa, including microbes, insects, and vertebrates. Moreover, kin selection has shed light on a range of biological phenomena, including dispersal, sex-ratio adjustment, worker–queen conflicts in insect colonies, the distribution of reproduction in animal societies (reproductive skew), parasite virulence, genomic imprinting, and the evolution of multicellularity (Bourke 2011a). The principles of kin selection also help illuminate aspects of the major transitions in evolution, which occur when free-living individuals coalesce to form a new higher-level entity that eventually becomes an individual itself (Maynard Smith and Szathmary 1995, Bourke 2011a).

Despite its empirical success, kin selection theory is not without its critics. For example, E. O. Wilson, the famous author of Sociobiology, was once an enthusiastic supporter of kin selection but has since changed his mind. In their recent work on eusocial insect colonies, Wilson and his coauthor Bert Hölldobler (Wilson and Hölldobler 2005, Hölldobler and Wilson 2008) argued that genetic relatedness is less important than is often thought; according to their view, ecological factors, rather than high levels of within-colony relatedness, are the primary drivers of the evolution of eusociality.

In August 2010, a strongly worded critique of kin selection by Nowak, Tarnita, and Wilson (2010) ignited a new round of debate in Nature. In March 2011, a rebuttal was published, signed by 137 social evolution theorists who claimed that Nowak and colleagues’ (2010) arguments “are based on a misunderstanding of evolutionary theory and a misrepresentation of the empirical literature” (Abbot et al. 2011, p. E1). More-detailed rebuttals have since appeared (Gardner et al. 2011, Bourke 2011b, Rousset and Lion 2011), as has a response by Nowak and colleagues (2011). Follow-up critiques by van Veelen and colleagues (2012), Wilson (2012), Allen and colleagues (2013), and Wilson and Nowak (2014) have left continuing uncertainty about the status of Hamilton's (1964) theory. Does it lie in tatters, or is it alive and kicking, healthier than ever? It depends on whom you ask.

In this overview, we offer a fresh look at some of the issues raised by this debate. As philosophers of science, rather than practicing biologists, we hope to bring a certain detachment to the discussion. Our aim is not to debunk or vindicate kin selection, nor to take a stand on any empirical questions, but to offer some conceptual clarifications. In the next section, we discuss the core explanatory principle of kin selection theory, Hamilton's rule. We emphasize that, although the name suggests a single unambiguous principle, there are, in fact, various formulations of the rule that it is crucial to distinguish. In the following section, we examine the relationship between the neighbor-modulated fitness and inclusive fitness approaches to kin selection and look briefly at the idea that inclusive fitness is the quantity that organisms should appear designed to maximize. Then we examine the often strained relationship between the theories of kin and group selection and ask whether these theories are ultimately equivalent, as is often claimed. In the final section, we close by highlighting some outstanding issues.

## Key issue 1: The status of Hamilton's rule

The central explanatory principle of kin selection theory is Hamilton's rule, which says that a gene coding for a social behavior will be favored by natural selection if and only if rb > c, where b represents the benefit that the behavior confers on the recipient, c represents the cost that it imposes on the actor, and r is the coefficient of relatedness between the actor and the recipient (Hamilton 1964). The costs and benefits are measured in units of reproductive fitness. The rule tells us that an altruistic behavior will be favored by selection so long as the fitness cost to the actor is offset by a sufficient amount of benefit to sufficiently closely related recipients.

In contemporary discussions, r is intended to encompass any relevant genetic similarity between actors and recipients, regardless of the mechanism that led to it. Therefore, although Hamilton (1964) originally defined r in genealogical terms—as a measure of shared ancestry—in principle, Hamilton's rule still applies when genetic correlations arise by other means, including green-beard effects (Dawkins 1976, Gardner and West 2010); pleiotropic effects (Hamilton 1975); and, in microbes, gene mobility (Mc Ginty et al. 2013, Birch 2014a). In practice, however, genealogical kinship remains the most common source of genetic correlation between social partners.

Nowak and colleagues (2010) stated that Hamilton's rule “almost never holds” (p. 1059), in the sense that it almost never constitutes a true statement of the conditions under which a social behavior will be favored by natural selection. This claim elicited vigorous rebuttals from their opponents—most notably from Gardner and colleagues (2011), who retorted that “it is simply incorrect to claim that Hamilton's rule requires restrictive assumptions or that it almost never holds” (p. 1038). There is, at present, no sign of an end to this divisive dispute (see Nowak et al. 2011, Allen et al. 2013, West and Gardner 2013). It is hard to see how both camps can be right, but neither seems likely to budge.

## Three versions of Hamilton's rule: Special, general, and approximate

The key to understanding the current standoff is to see that, when social evolution theorists talk about Hamilton's rule, they may have a number of subtly different principles in mind. Hamilton (1964) first derived a result of the form rb > c in a one-locus population-genetic model in which a number of substantial assumptions were made, including weak selection, additive gene action (i.e., no dominance or epistasis), and the additivity of fitness payoffs (i.e., a relatively simple payoff structure). In the following decades, numerous theorists (including Hamilton himself) explored the extent to which a similar result could be recovered when some or all of Hamilton's (1964) original assumptions were relaxed. The upshot was a variety of different routes to rb > c–type results, often with contrasting implications about the conditions under which the rule applies (e.g., Hamilton 1975, Michod 1982, Queller 1984, 1992, Grafen 1985, Frank 1998, 2013, Rousset 2004, Lehmann and Keller 2006, Lehmann and Rousset 2010, 2014a, 2014b).

Within this rather bewildering space of alternative formulations of Hamilton's rule, one three-way distinction is particularly salient. It concerns the meanings of the cost and benefit coefficients. First of all, there are formulations in which cost and benefit denote the payoff parameters of a specific evolutionary model—for example, the formulations of Queller (1984), Taylor C and Nowak (2007), van Veelen (2009), Nowak and colleagues (2010), and van Veelen and colleagues (2012). Second, there are formulations in which the cost and benefit terms are partial regression coefficients (i.e., average effects, in the sense of Fisher 1941) that quantify the overall statistical associations in a population among an organism's genotype or phenotype, its fitness, and the genotype or phenotype of social partners—which can, in principle, be computed for any model or set of population data. Queller's (1992) formulation is one example, recently defended and applied by Gardner and colleagues (2007, 2011). Third, there are formulations in which cost and benefit refer to marginal, first-order approximations of regression coefficients. This is the approach most commonly used by contemporary kin selection theorists. Roughly speaking (because this is not the place for detailed mathematical exposition), the approximation works by replacing differences with differentials. That is, it approximates the regression coefficients corresponding to c and b with partial derivatives of a fitness function (Taylor PD and Frank 1996, Frank 1998, 2013, Rousset 2004, Lehmann and Rousset 2010, 2014a, 2014b).

Some clear labels will help us keep these versions apart. For the exact version of the rule in which c and b are payoff parameters, we suggest the abbreviation HRS (the S is for special). For the exact, regression-based version of Queller (1992), we suggest the abbreviation HRG (G for general). For the marginal approximation of HRG, we suggest the abbreviation HRA (A for approximate).

Which version we have in mind makes an important difference to the generality of Hamilton's rule. HRS is an exact result for any model with an additive payoff structure—that is, a payoff structure in which the payoff that an actor's behavior confers on a recipient is independent of the recipient's phenotype and combines with other payoffs by adding up. This, however, amounts to a significant restriction. It is easy to construct counterexamples to HRS simply by considering a nonadditive payoff structure in which the payoff that a given social action confers on a recipient does depend on the recipient's own phenotype. This point was noted by Queller (1984) and was recently emphasized by van Veelen (2009). Unsurprisingly, when the payoff structure of social interaction is too complex to be represented with just two parameters (as is the case in nonadditive scenarios), a rule more complicated than HRS is needed to describe the condition for a social behavior to spread (Queller 1984, van Veelen 2009).

However, if we define c and b as partial regression coefficients (as in HRG), we obtain a version of Hamilton's rule of much greater generality. Indeed, we end up with an exact version of the rule that remains correct no matter how complicated the payoff structure may be, because all relevant payoff parameters are implicitly taken into account in the calculation of the costs and benefits (Queller 1992, Gardner et al. 2007, 2011). In effect, this is because we are abstracting away from the complex causal details of social interaction to focus on the overarching statistical relationship between genotype and fitness. This generalized, regression-based version of Hamilton's rule is always true because it makes no assumptions at all about how these statistical relationships are mediated phenotypically.

The marginal approximation of HRG (i.e., HRA) sacrifices a degree of this generality, because the approximation of differences by differentials is justified only if selection is weak and gene action is additive (Frank 1998, Lehmann and Rousset 2014b). However, HRA does not presuppose an additive payoff structure, and it therefore holds (unlike HRS) across a wide range of game-theoretic scenarios. The key is that HRA is fundamentally an approximate result. Rather than assuming that the payoff structure is additive, HRA relies on the idea that, when selection is weak, a first-order approximation that neglects deviations from payoff additivity is justified. In broad terms, then, HRA provides an intermediate degree of generality. Its assumptions are more restrictive than those of HRG but less restrictive than those of HRS.

We can use this three-way distinction to make sense of the ongoing standoff. When Nowak and colleagues (2010) said that “Hamilton's rule almost never holds,” they were referring to HRS, the exact version of the rule in which c and b refer to payoff parameters. Meanwhile, when Gardner and colleagues (2011) said that “it is simply incorrect to say that Hamilton's rule requires restrictive assumptions or almost never holds,” they were referring to the exact, regression-based version employed by Queller (1992), Gardner and colleagues (2007), and others. Once we distinguish HRS from HRG, we see that both of these apparently contradictory statements are correct (Birch 2014b). Neither statement here is referring to HRA, even though this approximate version of the rule is the version most commonly used by kin selection theorists.

## Does HRG explain anything?

Getting clear about the definitions of cost and benefit does not wholly resolve the conflict over Hamilton's rule, because, underneath the terminological fog of war, there are substantive issues at stake. One question is whether, if (as in HRG) we define the c and b terms so that Hamilton's rule is always true, we buy generality at the cost of explanatory power. As far as Nowak and colleagues (2011) are concerned, HRG adds nothing to our understanding of social evolution:

There are attempts to make Hamilton's rule work by choosing generalized cost and benefit parameters [HRG], but these parameters are no longer properties of individual phenotypes. They depend on the entire system, including population structure. These extended versions of Hamilton's rule have no explanatory power for theory or experiment. (Nowak et al. 2011, p. E9)

Do Nowak and colleagues (2011) have a case? It is undoubtedly true that HRG has predictive limitations. For example, one might expect Hamilton's rule to predict that if we were to intervene to increase the genetic relatedness between social partners, cooperative behavior would be more likely to evolve. But there are simple models in which the r, c, and b coefficients in HRG are all interdependent, with the result that intervening to increase relatedness also increases the cost:benefit ratio, which, in turn, makes cooperative behavior less likely to evolve. Similarly, one might intuitively predict that if a social behavior satisfies Hamilton's rule at one time, it will continue to do so in the future, provided there is no change in the underlying payoff structure or the relatedness between social partners. But the c and b coefficients in HRG will typically depend on population gene frequency, with the consequence that a social behavior may satisfy HRG at a low frequency but not at a higher frequency (Allen et al. 2013, Birch 2014b, Lehmann and Rousset 2014a).

These concerns about the predictive limitations of HRG are real but do not imply that it has no explanatory power at all. This is because, although prediction and explanation are related, they are not exactly the same thing. As philosophers of science have often noted, a principle can be explanatory without being predictive and vice versa (Salmon 1989). In the philosophy of science, there is a long tradition of pointing to unification as an important aspect of scientific explanation (Kitcher 1989). In this spirit, some defenders of HRG have argued that it constitutes a unifying principle in social evolution theory that helps us see what otherwise disparate models have in common (Gardner et al. 2007, Birch 2014b).

However, in addition to its unifying power, Hamilton's rule is often also taken to embody an important causal insight about social evolution—namely, that a costly social behavior will spread only if the direct fitness effect of the behavior on the actor who performs it is outweighed by the indirect fitness effect on the recipient, weighted by the relatedness between them, where effect is understood causally and not just statistically. This causal interpretation of HRG is valid only if the c and b regression coefficients allow an interpretation as causal effects. It is not entirely clear when it is legitimate to interpret them in this way, because there is no general theory of when, exactly, a partial regression coefficient (or Fisherian average effect; Fisher 1941) admits of a causal interpretation. The debate is ongoing, and connects in interesting ways to debates surrounding Fisher's fundamental theorem (Lee and Chow 2013). What we do know, however, is that partial regression coefficients are certainly not causally interpretable in all cases (Spirtes et al. 2000, Queller 2011, Allen et al. 2013, Birch 2014b). To think otherwise is to confuse causation and correlation. Indeed, Allen and colleagues (2013) provided several hypothetical examples in which a causal interpretation of the coefficients is not reasonable.

By this point, it is clear that the debate has taken on a partly philosophical character, turning on subtle issues concerning the relationship between causality and statistics and the explanatory function that Hamilton's rule is intended to serve. These are issues that neither mathematical modeling nor empirical studies can decisively settle. For this reason, debates about the value of HRG are unlikely to go away. But if researchers manage to steer clear of semantic confusions fostered by the alternative formulations of Hamilton's rule, there is room for a constructive debate regarding the rule's explanatory uses and limitations.

## Key issue 2: The status of inclusive fitness

Hamilton's (1964) original paper introduced the concept of inclusive fitness, a modification of the classical fitness concept for dealing with social interactions. An organism's inclusive fitness is defined as a weighted sum, over all individuals in the population (including itself), of those portions of each individual's reproductive output for which the organism is causally responsible, with the weights given by relatedness coefficients. Hamilton observed that an altruistic action, which, by definition, will reduce an organism's personal fitness, may nonetheless enhance its inclusive fitness, and he proposed that social evolution be understood as a process of inclusive fitness maximization. The status of the inclusive fitness concept is another bone of contention in the current controversy. Nowak and colleagues (2010) and Allen and colleagues (2013) have argued that the concept has no advantages over the traditional fitness concept. By contrast, Grafen (2006), Bourke (2011a), and West and Gardner (2013) have argued that inclusive fitness is the key to understanding social evolution.

## Neighbor-modulated and inclusive fitness

Inclusive fitness is not the only way to formulate kin selection theory. As Hamilton (1964) himself noted, an alternative is to use neighbor-modulated fitness, which is, in some ways, a more intuitive notion. To see the difference between them, consider two viewpoints on what happens when altruism evolves by virtue of relatedness between social partners (figure 1a, 1b). One viewpoint is that relatedness is a source of correlated interaction: When the value of r is high, bearers of the genes for altruism are differentially likely to interact with other bearers and, therefore, to receive the benefits of other agents’ altruism. The upshot is that, a high r value means that bearers of the genes for altruism may have greater reproductive success, on average, than nonbearers. The other is to view relatedness as a source of indirect reproduction: When the value of r is high, recipients provide actors with an indirect means of securing genetic representation in the next generation. Therefore, the genes for altruism may spread, if the indirect representation that an altruist secures through helping its relatives exceeds the representation that it loses through sacrificing a portion of its own reproduction success.

Figure 1a.

Relatedness leads to correlated interaction. Two altruists (black) confer a fitness benefit (b) on each other at a cost (c) to themselves. As a result, they are fitter overall than two nearby nonaltruists (white). Genetic relatedness can give rise to such patterns of correlated interaction in a population, making altruists fitter (on average) than nonaltruists.

Figure 1a.

Relatedness leads to correlated interaction. Two altruists (black) confer a fitness benefit (b) on each other at a cost (c) to themselves. As a result, they are fitter overall than two nearby nonaltruists (white). Genetic relatedness can give rise to such patterns of correlated interaction in a population, making altruists fitter (on average) than nonaltruists.

Figure 1b.

Relatedness leads to indirect reproduction. An altruist (black) confers a fitness benefit (b) on a related recipient (white) at a cost (c) to itself. The recipient does not express the altruistic phenotype. However, it possesses conditionally expressed genes for altruism, which it transmits to some of its offspring (indicated by the dotted lines, which show the genetic similarity between the actor and the recipient's offspring). The recipient thereby provides the actor with a means of indirect reproduction—that is, an indirect route to genetic representation in the next generation.

Figure 1b.

Relatedness leads to indirect reproduction. An altruist (black) confers a fitness benefit (b) on a related recipient (white) at a cost (c) to itself. The recipient does not express the altruistic phenotype. However, it possesses conditionally expressed genes for altruism, which it transmits to some of its offspring (indicated by the dotted lines, which show the genetic similarity between the actor and the recipient's offspring). The recipient thereby provides the actor with a means of indirect reproduction—that is, an indirect route to genetic representation in the next generation.

The first perspective is captured in the neighbor-modulated fitness framework (figure 2), which looks at the correlations between an individual's genotype and its social neighborhood and helps predict when these correlations will make the bearers of the genes for altruism fitter, on average, than nonbearers (Hamilton 1964, Taylor PD and Frank 1996, Frank 1998, 2013). The second perspective is captured in the inclusive fitness framework (figure 3), which adds up all the fitness effects causally attributable to a social actor, weighting each component by the relatedness between the actor and the recipient, in order to calculate the net effect of a social behavior on the actor's overall genetic representation in the next generation (Hamilton 1964, Frank 1998, 2013, Grafen 2006).

Figure 2.

Neighbor-modulated fitness. In a neighbor-modulated fitness analysis, we ascribe to A those fitness components that correspond to its personal reproductive success. Some of these components are influenced by the behavior of B, C, and D (as is shown by the arrows). A's total neighbor-modulated fitness is a simple sum of these components (3b), plus a component corresponding to A's own influence on its reproductive success (–c), plus a baseline component independent of the character of interest.

Figure 2.

Neighbor-modulated fitness. In a neighbor-modulated fitness analysis, we ascribe to A those fitness components that correspond to its personal reproductive success. Some of these components are influenced by the behavior of B, C, and D (as is shown by the arrows). A's total neighbor-modulated fitness is a simple sum of these components (3b), plus a component corresponding to A's own influence on its reproductive success (–c), plus a baseline component independent of the character of interest.

Figure 3.

Inclusive fitness. In an inclusive fitness analysis, fitness effects are assigned to the actors whose behavior was causally responsible for them. A therefore retains the effect –c for which it responsible but loses the 3b units of personal fitness it received by virtue of its interactions with B, C, and D. In compensation, it gains 3b units taken from the reproductive output of B, C, and D. To calculate A's inclusive fitness, these new slices are weighted by the actor's relatedness to the recipient.

Figure 3.

Inclusive fitness. In an inclusive fitness analysis, fitness effects are assigned to the actors whose behavior was causally responsible for them. A therefore retains the effect –c for which it responsible but loses the 3b units of personal fitness it received by virtue of its interactions with B, C, and D. In compensation, it gains 3b units taken from the reproductive output of B, C, and D. To calculate A's inclusive fitness, these new slices are weighted by the actor's relatedness to the recipient.

Although correlated interaction and indirect reproduction may sound like different mechanisms, the inclusive and neighbor-modulated fitness frameworks are usually considered equivalent, because they generally yield identical results about when a social behavior will evolve (Taylor PD et al. 2007). Therefore, the choice is one of modeling convenience, not empirical fact. Hamilton (1964) and Maynard Smith (1983) both regarded inclusive fitness as easier to apply in practice, but, in recent years, this situation has largely reversed: Kin selection theorists have increasingly come to favor the neighbor-modulated fitness framework, citing its greater simplicity and ease of application (Taylor PD and Frank 1996, Gardner et al. 2007, Taylor PD et al. 2007).

In one respect, the neighbor-modulated approach is more general. To perform an inclusive fitness analysis, we need to be able to attribute each social phenotype to a single controlling genotype (Frank 1998). By contrast, a neighbor-modulated fitness analysis simply ignores the pathway from actor genotypes to social phenotypes, leaving us with one fewer causal path to worry about. A corollary is that the neighbor-modulated framework can apply in cases in which there is no principled way to ascribe a social character to a single controlling genotype. As Frank (1998, 2013) noted, cases in which phenotypes are controlled by actors of a species different from that of the recipient—such as host–parasite interactions—arguably fall into this category (but cf. Taylor PD et al. 2007).

## Inclusive fitness and the objective of social behavior

One advantage of the inclusive fitness approach is that it helps to make precise the idea that an organism's social behavior is purposive, or goal oriented. This idea of purpose—or apparent purpose—is a key component of the adaptationist approach to evolution that Darwin initiated. When nonsocial traits are concerned, biologists typically assume that an evolved trait will serve to enhance an organism's expected reproductive output; models based on the assumption often enjoy empirical success. But altruistic behaviors seemingly do not fit this paradigm, because they reduce rather than enhance an organism's personal fitness. It is here that the inclusive fitness concept comes into its own, allowing us to rescue the idea that social behavior should appear purposive by suitably redefining the purpose in question—namely, the enhancement of inclusive rather than personal fitness. This feature of the inclusive fitness concept explains its popularity among behavioral ecologists and has been emphasized in recent work by Grafen (2006, 2014), Gardner and colleagues (2011), Okasha and colleagues (2014), and others.

What enables inclusive fitness to play this role is its focus on which actors control which phenotypes. Recall that an actor's inclusive fitness is a relatedness-weighted sum of the fitness effects for which it is causally responsible. Therefore, we can put ourselves in the position of the actor and ask, How should I behave in order to maximize my expected inclusive fitness? Because natural selection tends to favor traits that promote inclusive fitness on average, this question can serve as an informal route to predictions of which social behaviors will evolve. By contrast, we cannot usefully ask the same question with regard to neighbor-modulated fitness, because an individual's neighbor-modulated fitness contains components over which it may have no control. All we can do is put ourselves in the position of a recipient and ask, What genotypes are correlated with good outcomes as far as my neighbor-modulated fitness is concerned? But this heuristic is considerably less intuitive, because considerations of causation and control are replaced by considerations of statistical auspiciousness.

The idea that social behavior should serve to maximize an organism's inclusive fitness is hinted at in Hamilton's (1964) original paper but not made fully explicit. In his recent work on the formal Darwinism project, Grafen (2006, 2014) has attempted to place the idea on a firm footing, by proving formal links between gene-frequency change and an optimization program. Essentially, Grafen (2006, 2014) sought to prove, in a quite general setting, that if all the organisms in a population choose an action (from a fixed set of possible actions) that maximizes their inclusive fitness, population-genetic equilibrium will obtain and vice versa. Although (as Grafen 2006, 2014 admitted) this falls short of proving that natural selection will always lead inclusive fitness maximizing behavior to evolve (e.g., because gene frequencies may cycle indefinitely), it arguably provides some support for that belief. In effect, Grafen's (2006, 2014) results (taken at face value) mean that, so long as the population does actually evolve toward a stable equilibrium, we should expect inclusive-fitness maximizing behavior to evolve.

Grafen's (2006, 2014) results rest on one key assumption—namely, that costs and benefits have additive phenotypic effects on fitness. This means, for example, that the benefit b that an altruistic action has on the recipient is independent of the recipient's own genotype. In general, this is not a realistic assumption, because it rules out any frequency dependence of fitness, although it may be a good approximation in certain cases. Whether Grafen's (2006, 2014) results can be extended to the nonadditive case has not yet been settled (see Lehmann and Rousset 2014a, Gardner and colleagues 2011 for conflicting opinions on this issue).

At this point, it is useful to recall the general formulation of Hamilton's rule (HRG), which, as we saw, defines the r, b, and c coefficients in such a way that the rb > c condition is always correct, irrespective of whether costs and benefits are additive. It is tempting to suggest that Grafen's (2006, 2014) optimization results could be extended to the nonadditive case and, therefore, made fully general, simply by defining inclusive fitness using the r, b, and c terms of HRG. However, there is a problem with this suggestion; recall that an organism's inclusive fitness is supposed to be fully within its control (i.e., to depend only the social actions that it performs). Because the b and c terms of HRG are functions of population-wide gene frequencies, the amount of inclusive fitness that an organism gets from a given action would depend on the state of the population, if inclusive fitness were defined as we have suggested.

This suggests that the generalization of Grafen's (2006, 2014) results on inclusive fitness maximization to the nonadditive case will be difficult to achieve. Furthermore, it highlights the important difference between Hamilton's rule, itself—the statement of the conditions under which an allele for a social behavior will be favored by selection—and the idea that an organism's evolved behavior will serve to maximize its inclusive fitness. These two aspects of kin selection theory, although they are related, should be kept distinct.

## Key issue 3: Kin selection and multilevel selection

Another dimension of the current controversy concerns the relationship between kin and multilevel (or group) selection. Kin and multilevel selection provide seemingly quite different perspectives on social evolution. Kin selection, as we have seen, emphasizes the relatedness between social partners as the crucial factor mediating the spread of a prosocial behavior. Multilevel selection, in contrast, emphasizes the interplay of selection within groups and between groups (Price 1972, Hamilton 1975, Sober and Wilson 1998, Okasha 2006). Within any group, altruists will be at a selective disadvantage vis á vis their selfish counterparts, but groups containing a high proportion of altruists may outcompete groups containing a lower proportion. So, for an altruistic behavior to spread, the between-group component of selection must trump the within-group component.

The relationship between kin and multilevel selection has been a source of controversy ever since it was first broached by Hamilton (1975). In earlier debates, biologists tended to regard kin and multilevel selection as rival empirical hypotheses (e.g., Maynard Smith 1964, 1976, Dawkins 1976), but many contemporary biologists regard them as ultimately equivalent, on the grounds that gene frequency change can be correctly computed using either approach (e.g., Lehmann et al. 2007, Marshall 2011, Frank 2013). Although dissenters from this equivalence claim can be found (e.g., Hölldobler and Wilson 2009, van Veelen 2009, Nowak et al. 2010, Traulsen 2010), the majority of social evolutionists appear to endorse it.

## Formal equivalence

To see the grounds for the equivalence claim, consider a simple model. A population of haploid individuals live in groups of the same size, within which social interactions occur (figure 4). An allele at a particular locus codes for a social behavior. Define pi to be 1 if the ith individual has the allele and pi to be 0 otherwise. The index i ranges over all individuals in the global population, irrespective of group membership. The population-wide frequency of the allele is $${\bar p}$$. The reproductive output (i.e., fitness) of individual i, defined as the total number of surviving offspring that it contributes to the next generation, is denoted by wi. The average fitness in the population is $${\bar w}$$. Mutation is assumed absent.

Figure 4.

Individuals in a group-structured population. The black dots represent individuals with the allele of interest, the white dots represent nonbearers, and the larger circles denote social groups.

Figure 4.

Individuals in a group-structured population. The black dots represent individuals with the allele of interest, the white dots represent nonbearers, and the larger circles denote social groups.

Under these assumptions, the change in allele frequency over a single generation is given by

(1)
$$\bar w\Delta \bar p = {\mathop{\rm cov}} (w_i ,p_i ).$$
This is a version of the Price equation (Price 1970); the full version includes an extra term, but we are entitled to drop that term here because our assumptions guarantee the unbiased transmission of alleles. The equation tells us that the allele—and, therefore, the social behavior that it codes for—will spread so long as cov(wi, pi) is greater than 0 (i.e., there is a positive covariance between an individual's fitness and its genetic value). This simply formalizes the core neo-Darwinian idea that genes associated with higher individual fitness will increase in frequency.

Equation 1 is always true but not always useful, becauase the covariance term will often lack a natural biological interpretation (Grafen 2006, Okasha 2014). Kin and multilevel selection can be regarded as alternative ways of decomposing the covariance term in equation 1 into more meaningful components. According to the kin selection approach, we use a linear regression model to split the covariance term into components attributable to the direct and indirect fitness effects of the social behavior under consideration (Queller 1992, Gardner et al. 2011). This allows us to derive HRG, the generalized version of Hamilton's rule discussed above, in a straightforward manner. According to the multilevel selection approach, we split the covariance term into components attributable to selection within groups and selection between groups (Price 1972, Okasha 2006). This allows us to derive a principle that closely parallels HRG, according to which a costly social behavior can spread by natural selection only if the selection for the trait between groups is stronger than the selection against the trait within groups. The details of these derivations are spelled out in boxes 1 and 2.

Box 1. Kin selection approach.

If wi is the fitness of individual i, pi is the genetic value of individual i, pi is the average genetic value of individual i's social partners, we can write wi as a multiple regression on pi and p′i:

(2)
$$w_i = \alpha + \beta _{wp.p'} p_i + \beta _{wp'.p} p_I + e_i .$$

Substitute equation 2 into 1 to yield

(3)
$$\bar w\Delta \bar p = (\beta _{wp.p'} + \beta _{wp'.p} \beta _{p'p} ){\mathop{\rm var}} (p),$$
where βpp is the linear regression of p′ on p.

Relabel βwp.p′ and βwp.p as –c and b, respectively, and βpp as r, to give

(4)
$$\bar w\Delta \bar p = \overbrace {( - c){\mathop{\rm var}} (p)}^{{\rm direct}\,{\rm effect}} + \overbrace {(rb){\mathop{\rm var}} (p)}^{{\rm indirect}\,{\rm effect}}.$$

Equation 4 yields the generalized Hamilton's rule:$$\Delta \bar p > 0$$ if and only if rb > c (provided that var(p) ≠ 0).

Box 2. Multilevel selection approach.

If pjk is the genetic value of the jth individual in the kth group, wjk is the fitness of the jth individual in the kth group, Pk is the average genetic value of the kth group, and Wk is the average fitness of the kth group, the overall covariance between w and p, in the global population, can be written as

(5)
$${\mathop{\rm cov}} (w_i ,p_i ) = \overbrace {{\mathop{\rm cov}} (W_k ,P_k )}^{{\rm between - group}} + \overbrace {E_k [{\mathop{\rm cov}} (w_{jk} ,p_{jk} )]}^{{\rm within - group}},$$
where cov(Wk,Pk) is the covariance between the group means and Ek[cov(wjk, pjk)] is the average of the within-group covariances between w and p.

Substituting equation 5 into equation 1 yields the following:

(6)
$$\bar w\Delta \bar p = \overbrace {{\mathop{\rm cov}} (W_k ,P_k )}^{{\rm between - group}} + \overbrace {E_k [{\mathop{\rm cov}} (w_{jk} ,p_{jk} )]}^{{\rm within - group}}.$$

Equation 6 tells us that

\begin{equation*} \Delta \bar p > 0\,{\rm if}\,{\rm and}\,{\rm only}\,{\rm if}\,{\mathop{\rm cov}} (W_k ,P_k ) > - E_k [{\mathop{\rm cov}} (w_{jk} ,p_{jk} )]. \end{equation*}

We can now see why kin and multilevel selection are often regarded as equivalent. In any group-structured population, the total evolutionary change can be decomposed using either the kin selection partition (equation 4) or the multilevel partition (equation 6). Moreover, it is easy to see that the kin selection criterion for the spread of a prosocial trait (rb > c) will be satisfied if and only if the multilevel criterion is satisfied (i.e., the covariance between groups is greater than that within groups). Therefore, the two approaches are formally equivalent. Gene frequency change can be computed in two ways: by determining the magnitude of the between- and within-group components or determining that of the direct and indirect effects; the two methods will always give the same answer. In effect, the two approaches can be seen as alternative ways of capturing the fundamental insight that positive assortment (i.e., altruists interacting preferentially with each other) is what is crucially needed for altruism to evolve.

Recently, van Veelen (2009) and van Veelen and colleagues (2012) challenged the received wisdom on this issue, arguing that kin and multilevel selection are not formally equivalent and that the latter is, in fact, more general than the former (see also Traulsen 2010). The HRS–HRG distinction introduced above again helps us understand what is going on here. What van Veelen and colleagues (2012) showed, in effect, is that the special version of Hamilton's rule, HRS, is not formally equivalent to the standard multilevel decomposition in box 2. This is true but should come as no surprise, because HRS applies only under restrictive assumptions. Their argument does not threaten the equivalence results of Marshall (2011) and others, because these results concern the formal equivalence of the multilevel selection approach and the general version of Hamilton's rule. Again, the key is to distinguish between the maximally general formulation of kin selection (i.e., HRG) and more specific formulations.

In one respect, the kin selection approach is arguably more general than the multilevel approach, because the latter requires that individuals be nested into nonoverlapping groups, as in figure 4; this is necessary for the decomposition technique in box 2 to apply (Hamilton 1975, Okasha 2006, Frank 2013). Groups of this sort exist in some taxa (e.g., the colonies of many social insect species). But in other cases, individuals engage in social interactions with their conspecifics, but there are no well-defined, discrete groups. The kin selection approach can handle such cases easily; indicative of this is that in deriving equation 4 above (box 1), we did not make use of the fact that the individuals were nested into nonoverlapping groups. Therfore, the claim that kin and multilevel selection are formally equivalent requires at least this qualification.

## Choosing between them

On a practical, day-to-day basis, social evolution researchers must decide which approach to use, and the formal equivalence of the two approaches does not imply that there is no principled basis on which to choose between them. West and colleagues (2008) were emphatic on this point:

At one level, kin selection and group selection are just different ways of doing the math or conceptualizing the evolutionary process. However, from a practical point of view, it could not be clearer that the kin selection approach is the more broadly applicable tool that we can use to understand the natural world. This is because kin selection methodologies are usually easier to use, allow the construction of models that can be better linked to specific biological examples, lend themselves to empirical testing, and allow the construction of a general conceptual overview. In addition, the group selection approach is not only less useful but also appears to frequently have negative consequences by fostering confusion that leads to wasted effort (West et al. 2008, pp. 381–382).

Is this a fair assessment? It is true that the kin selection approach (in both its neighbor-modulated and inclusive fitness guises) has received more theoretical attention than the group selection approach and has been put to work in more empirical applications. For example, kin selection models can straightforwardly take into account class structure, whereby different types of social agent in a population have different reproductive value (Taylor PD 1990, Frank 1998), and they are readily used in conjunction with the Taylor–Frank method, a powerful technique for the prediction of evolutionarily stable strategies (Taylor PD and Frank 1996, Frank 1998). However, this does not show that the multilevel approach is unworthy of a similar degree of theoretical attention or that it is inherently unsuited to empirical applications. Indeed, given that kin and multilevel selection are formally rather similar (they simply partition up the total evolutionary change in slightly different ways), claims that one approach is inherently superior to the latter, as proponents of each have argued, must be treated with a degree of skepticism.

The widespread preference for kin selection may be partly due to multilevel selection's association with the flawed good-of-the-group tradition of the 1950s and 1960s and the associated superorganism concept, of which many biologists remain suspicious. It is undeniable that the careless appeal to group-level advantage as a way of explaining a trait's evolution led to serious errors in the past, so biologists’ wariness of this mode of explanation is understandable. Kin selection is an individualistic methodology that makes no explicit mention of group fitness or group advantage, so it has often seemed preferable for that reason (e.g., Dawkins 1976). However, this consideration should not be overplayed. Past errors not notwithstanding, multilevel selection has evolved into a respectable theory and does not necessarily carry a commitment to the superorganism concept (which is, at best, defensible only in special cases, such as clonal groups or highly advanced eusocial insect colonies; cf. Gardner and Grafen 2009, Okasha and Paternotte 2012). Moreover, the idea that kin selection is methodologically preferable to multilevel selection seems hard to square with their formal equivalence. Indeed, those who have favored kin selection on these grounds have typically not properly appreciated that equivalence (West et al. 2008 is an exception in this respect).

It has recently been suggested that kin selection has a unique advantage over multilevel selection, in that it comes with an associated optimization principle (Gardner and Grafen 2009, Gardner et al. 2011). The suggestion here is that the concept of organisms’ maximizing their inclusive fitness, which permits social behavior to be brought within the Darwinian paradigm, is the key insight of kin selection theory but has no good parallel in multilevel selection theory. The putative parallel would presumably involve groups maximizing their group fitness, but this notion only makes sense for fully clonal groups, it has been argued (Gardner and Grafen 2009, but cf. Okasha and Patternote 2012). This line of argument is interesting but not conclusive, given that the circumstances in which it has been shown that evolution will lead individuals to maximize their inclusive fitness are fairly restricted, as was emphasized above.

## Causal aptness

Finally, we want to suggest a different sort of consideration that might help biologists choose between the kin and multilevel approaches in a given context. The basic thought is that, although kin and multilevel selection are equivalent as statistical decompositions of evolutionary change, there are situations in which one approach provides a more accurate representation of the causal structure of social interaction. Evolutionary biology, as are other sciences, is interested in constructing causal explanations; ideally, we want our descriptions of evolutionary change to capture the causal structure of the underlying selection process, as well as correctly computing allele frequency change. So, although kin and multilevel selection may be formally equivalent, it does not follow that they are also equally good as causal representations.

For example, suppose we are investigating a segregation distorter allele that also has deleterious effects on the fitness of its bearer. It is very natural to describe the selection pressures operating on this allele in multilevel terms: At the gene level, there is selection in its favor, but at the organism level, there is selection against it. The formal equivalence of kin and group selection suggests that, if we wanted, we could redescribe the entire situation in terms of the inclusive fitness interests of the allele, but it is not clear what we stand to gain in explanatory terms by doing so. On the contrary, this move would seem unhelpful: It would obscure the true causal structure of the scenario, which clearly involves two distinct levels of selection. When we are looking at selection occurring both between and within organisms, a multilevel description seems clearly more apt, causally speaking.

However, there are other cases in which a kin selection description seems more apt from a causal point of view. Consider a prisoner's dilemma–style scenario in which organisms interact in pairs and must choose whether to cooperate or to defect. Suppose that genetic correlation between social partners leads to the evolution of cooperation. It seems natural to describe this in terms of kin selection: to say, for example, that organisms cooperate because it is in their inclusive fitness interests to do so. As Sober and Wilson (1998) pointed out, however, any such scenario may be redescribed in the language of multilevel selection, because if we regard each interacting pair as a group of size 2, we can say that, within each group, defectors outperform cooperators but that groups with more cooperators outperform groups with fewer. However, as in the previous example, it is not clear what we stand to gain from this rather strained description of the process. After all, these groups of size 2 may be highly ephemeral, coming into existence when the social interaction begins and vanishing as soon as it is complete. If this is the case, they are groups in name only, and describing this as a process of multilevel selection seems to sow confusion rather than insight.

Plainly, our intuitions about these two examples do not constitute a full-blown theory of causal aptness; they do not provide any general recipe for deciding which description is causally superior in any given case. Nevertheless, they are enough to show that considerations of causal aptness do matter, if we want our theories and models of social evolution to embody causal—as opposed to merely statistical—truths. Developing a more adequate treatment of causal aptness remains an important direction for future work. Okasha (2014) has attempted a systematic analysis of the circumstances under which kin and multilevel selection offer better causal representations of social evolution, using tools from the theory of causal modeling (Pearl 2009).

## Conclusions and open questions

There are many oustanding issues in the foundations of social evolution theory. We feel that progress on these issues is achievable if rival camps of researchers are able to communicate and cooperate, rather than pursuing divergent research programs. In this overview, we have tried to take an even-handed approach that identifies what both critics and defenders of kin selection have got right, while highlighting the ways in which theorists have, at times, talked past one another. We will close by highlighting three questions that we hope future work in this area will address.

## Question 1: When do the c and b coefficients in HRG admit of a causal interpretation?

Above, we noted that the generalized version of Hamilton's rule, HRG, defines the c and b coefficients using the statistical concept of regression. In effect, in applying HRG, we are fitting a plane to a three-dimensional cloud of population data describing each organism's genotype, its social partner's genotype, and its fitness; c and b are the coefficients that specify that plane. But can HRG tell us anything about the causal processes involved in the evolution of social behavior, given that it is defined in purely statistical terms? As Allen and colleagues (2013) have pointed out, following Spirtes and colleagues (2000), there are many cases in which regression coefficients should not be interpreted causally. The issue lies at the heart of the ongoing debates surrounding Hamilton's rule, but a systematic treatment is currently lacking.

## Question 2: How widely applicable is the idea that evolution will lead individuals to try to maximize their inclusive fitness?

We also noted that inclusive fitness appears to offer an objective for social behavior, because it is a quantity that is within the control of the individual actor. However, the most careful attempt to justify the idea that evolution in social contexts will lead individuals to behave as if they were trying to maximize their inclusive fitness, made by Grafen (2006), rests on assumptions that severely limit its generality. It is currently unclear whether Grafen's (2006) argument, or one like it, can be extended to cover nonadditive scenarios and to cover frequency-dependent selection.

## Question 3: Under what conditions are kin and multilevel selection causally, as opposed to formally, equivalent?

Finally, we also noted that kin and multilevel selection, when they are formulated in general terms as alternative decompositions of the Price equation, are formally equivalent, in that allele frequency change can be correctly computed in both ways. But intuitively, there are cases in which one is more causally apt than the other. However, a general account of causal aptness that goes beyond our intuitions in simple cases has yet to be constructed.

SO was supported by the European Research Council Seventh Framework Program (project no. FP7/20072013), grant agreement no. 295449.

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